More surprisingly, when the \(\mathcal {NC}\)\(_1\)  evaluated on the feature \(\mathbf {h}^\ell = \sigma { \mathbf {W}_{\ell -1} \cdots \sigma {\mathbf {W}_1 \mathbf {x} + \mathbf {b}_1} + \mathbf {b}_{\ell -1} }\)  across all intermediate layers \((1\le \ell \le L-1)\) , it can be further shown that it progressively decays from shallow to deep layers of a well-trained overparamterized deep neural network [1]}, [2]}, [3]}, as we illustrate in fig:res18nclayerwisedynamic. Similar to \(\mathcal {NC}\) , the layer-wise progressive collapse is universal, where it is also prevalent across the choice of training losses, network architectures, and dataset. Moreover, the recent work [4]} showed that the decay rate could be linear for simple networks such as MLPs, VGGs, and ResNet. The progressive decay of feature diversity implies that the deep network is discarding information of the input data from shallow to deep layers. In the following section, we will show that this observation would be very useful for explaining heuristic methods in pre-training and designing more efficient fine-tuning methods in a principled manner.
